#7622728

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁵
commutator, socle	C₂²	C₂⁴×C₈
prime radical	#3475	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1313 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(2,5)(3,7)(9,10,12,17,13,18,22,30,14,15,23,26,24,27,34,37)(11,16,20,28,21,29,33,38,19,25,31,35,32,36,39,40);(1,2)(3,6)(4,5)(7,8);(2,5)(6,8)(10,15)(16,25)(17,26)(18,27)(28,35)(29,36)(30,37)(38,40);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,20)(13,21)(14,19)(15,25)(17,28)(18,29)(22,33)(23,31)(24,32)(26,35)(27,36)(30,38)(34,39)(37,40);(11,19)(16,25)(20,31)(21,32)(28,35)(29,36)(33,39)(38,40)

Polycyclic presentation:

2^9|a^2=g,d^2=f,g^2=h,h^2=i,b^a=bf,c^a=ci,d^a=df,c^b=cf,e^d=ei

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529924);